Transcript Chapter 43 Molecules and Solids Dr. Jie Zou PHY 1371

Chapter 43
Molecules and Solids
Dr. Jie Zou
PHY 1371
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Outline
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Dr. Jie Zou
Molecular bonds
Bonding in solids
Energy states and spectra of
molecules
Free electron theory of metals
Band theory of solids
Electrical conduction in metals,
insulators, and semiconductors
Semiconductor devices and
superconductivity
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Molecular bonds
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The bonding mechanisms in a molecule are
fundamentally due to electric forces between
atoms (or ions).
Potential energy function that can be used to
model a molecule:
A B
U r    n  m for a two - atom system
r
r
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r: internuclear separation distance between the
two atoms
A and B are parameters that can be determined by
experiments.
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Plot of U(r) ~ r for a two-atom
system
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Equilibrium separation: U(r)
is a minimum and the two
atoms are in stable equilibrium.
At large separation distance: the
slope is positive, corresponding
to a net attractive force.
At small separation distance:
the slope is negative,
corresponding to a net
repulsive force.
Binding energy: The additional
energy the system has to be
given to break up the diatomic
molecule (so that r = ).
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Classification of molecular
bonding mechanisms
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Ionic bonds
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Covalent bonds
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Example: H2 molecule
Van der Waals bonds
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Example: Sodium Chloride (NaCl)
Example: Condensation of inert gas atoms
into the liquid phase
Hydrogen bonds
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Example: DNA molecules
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Example: Covalent bonding
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Ground-state wave functions
1(r) and 2(r) for two
hydrogen atoms making a
covalent bond ( 1s (r )  1 e r / a )
0
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Dr. Jie Zou
(a) The atoms are far apart
and their wave functions
overlap minimally.
(b) The atoms are close
together, forming a composite
wave function 1(r) + 2(r) for
the system. The probability
amplitude for an electron to be
between the atoms is high.
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Bonding in solids
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A Crystalline solid consists of a large
number of atoms (ions) arranged in a regular
array, forming a periodic structure.
Classification of bonding in solids:
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Ionic solids. Example: Sodium Chloride (NaCl
crystal)
Covalent solids. Example: Diamond, silicon,
germanium
Metallic solids. Example: Copper, silver, sodium,
etc.
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Examples of bonding in solids
NaCl
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Diamond
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Metal
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Energy states and spectra of
molecules
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Total energy of a molecule:
E = Eel + Etrans + Erot + Evib
Rotational motion of molecules
Vibrational motion of molecules
Molecular spectra
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Rotational motion of molecules
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Erot = (1/2) I2.
I = r2,  = m1m2/(m1 + m2), the
reduced mass of the molecule
Quantization of the magnitude of the
molecule’s angular momentum
L  I  J ( J  1) J  0, 1, 2,...
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J: rotational quantum number
Allowed 2values of rotational energy:
Erot 
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J ( J  1) J  0,1,2...
2I
Energy separation between adjacent
rotational levels: 2
E  E J  E J 1 
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4 I
2
J J  1,2,3...
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Allowed Rotational transitions
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Selection rule: J = 1
For most molecules, transitions
between adjacent rotational energy
levels result in radiation that lies in the
microwave range of frequencies (f ~
1011 Hz).
Example 43.1: The J = 0 to J = 1
rotational transition of the CO molecule
occurs at a frequency of 1.15 x 1011 Hz.
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Dr. Jie Zou
(a) Find the Moment of inertia of the
molecule.
(b) Find the bond length of the molecule.
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Vibrational motion of
molecules
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Frequency of vibration for the
system:
1 k
f 
2 
Allowed values of vibrational
energy:
Evib
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1
1 h
 (v  )hf  (v  )
2
2 2
k
v  0 ,1,2...
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Energy separation between
successive vibrational levels:
h k
Evib 
 hf
2 
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Allowed vibrational transitions
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Selection rule: v = 1
Transitions between vibrational
levels are caused by
absorption of photons in the
infrared region of the
spectrum.
Example 43.2: The frequency
of the photon that causes the
v = 0 to v = 1 transition in the
CO molecule is 6.42 x 1013 Hz.
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Dr. Jie Zou
Find the force constant k for this
molecule.
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Molecular spectra
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Total energy of the molecule:
1
2
E  (v  )hf 
J ( J  1)
2
2I
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Each energy level is indexed by the two
quantum numbers v and J.
Absorptive transitions between the v =
0 and v = 1 vibrational states:
(1) J = +1 and (2) J = -1
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Energies of the absorbed photons:
(1) E = hf + (ħ2/I)(J+1),
(2) E = hf - (ħ2/I)J,
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J= 0,1,2,…
J=1,2,3…
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Absorption spectrum of the
HCl molecule
Quick Quiz: There is a gap between the two sets of
peaks. Why?
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Homework
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Chapter 43, P. 1434, Problems: #3, 8,
9, 14, 17.
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